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Chapter 3: Problem 10

Solve each equation using the method of your choice. Then use a different method to verify your solution. a. \(14 x=63\) b. \(-4.5 x=18.6\) c. \(8=6+3 x\) d. \(5(x-7)=29\) e. \(3(x-5)+8=12\)

Short Answer

Expert verified
a) \(x=4.5\), b) \(x\approx-4.1333\), c) \(x=\frac{2}{3}\), d) \(x=\frac{64}{5}\), e) \(x=\frac{19}{3}\).

Step by step solution

01

Solving Equation a Using Division

To solve the equation \( 14x = 63 \), we first divide both sides by 14 to isolate \( x \). \[ x = \frac{63}{14} \]Performing the division gives us:\[ x = 4.5 \]
02

Verifying Equation a by Substitution

To verify the solution for \( 14x = 63 \), substitute \( x = 4.5 \) back into the original equation:\[ 14(4.5) = 63 \]Calculating this gives \( 63 = 63 \), which is true, thus confirming that \( x = 4.5 \) is correct.
03

Solving Equation b Using Division

To solve \(-4.5x = 18.6\), divide both sides by -4.5 to isolate \( x \).\[ x = \frac{18.6}{-4.5} \]Performing the division gives us:\[ x \approx -4.1333 \]
04

Verifying Equation b by Substitution

To verify the solution, substitute \( x \approx -4.1333 \) back into the equation:\[ -4.5(-4.1333) \approx 18.6 \]The left side simplifies to approximately 18.6, confirming the solution.
05

Solving Equation c Using Subtraction Then Division

First, subtract 6 from both sides of the equation \( 8 = 6 + 3x \):\[ 2 = 3x \]Now, divide both sides by 3 to isolate \( x \):\[ x = \frac{2}{3} \]
06

Verifying Equation c by Substitution

Substitute \( x = \frac{2}{3} \) back into the original equation:\[ 8 = 6 + 3\left(\frac{2}{3}\right) \]The equation simplifies to \( 8 = 8 \), confirming the solution is correct.
07

Solving Equation d by Distribution and Simplification

First, distribute the 5 on the left side of the equation \( 5(x - 7) = 29 \):\[ 5x - 35 = 29 \]Add 35 to both sides:\[ 5x = 64 \]Then divide by 5:\[ x = \frac{64}{5} \]
08

Verifying Equation d by Back Substitution

Substitute \( x = \frac{64}{5} \) back into the equation:\[ 5\left(\frac{64}{5} - 7\right) = 29 \]Simplify the left side to confirm it equals 29, verifying the solution.
09

Solving Equation e Using Distribution then Simplification

First, distribute the 3 in the equation \( 3(x - 5) + 8 = 12 \):\[ 3x - 15 + 8 = 12 \]Simplify:\[ 3x - 7 = 12 \]Add 7 to both sides:\[ 3x = 19 \]Divide by 3:\[ x = \frac{19}{3} \]
10

Verifying Equation e by Back Substitution

Substitute \( x = \frac{19}{3} \) back into the equation:\[ 3\left(\frac{19}{3} - 5\right) + 8 = 12 \]The simplified left side should equal 12, confirming the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Verification of Solutions
When you solve a linear equation, it's important to check if your solution is correct. This process is known as verification of solutions. Verification ensures that the solution you obtained actually satisfies the original equation.
To verify a solution:
  • Substitute the found value of the variable back into the original equation.
  • Perform all necessary arithmetic operations.
  • Check if both sides of the equation are equal; if they are, the solution is confirmed correct.
Verification is crucial because it helps identify errors before finalizing your answer. Always remember, a solved equation without verification might still lead to incorrect conclusions.
Equation Solving Methods
Solving linear equations involves finding the value of the variable that makes the equation true. There are various methods to solve these equations, each with its own advantages. Some common equation solving methods include:
  • Substitution Method: This involves replacing the variable in the equation with a known value to verify a solution.
  • Division Method: It involves isolating the variable by dividing both sides of the equation by a specific number.
  • Distribution: This is used for equations where terms need to be expanded before isolation.
Understanding these methods and when to apply them is key to efficiently solving linear equations. Each method can provide a different approach and insight into solving equations.
Substitution Method
The substitution method is a useful tool for solving and verifying solutions of equations. It involves substituting the value of the variable back into the equation to check if it solves the equation.
This method is particularly helpful during the verification process. Here's how it works:
  • After solving the equation and finding a solution for the variable, substitute this value into the original equation.
  • Perform the arithmetic calculations.
  • If both sides of the equation are equal after substitution, the solution is verified.
Substitution is a straightforward method that helps confirm whether the discovered value accurately solves the equation or not.
Division Method
The division method is frequently used to solve linear equations, particularly when we need to isolate the variable. It involves dividing both sides of the equation by the same non-zero number to solve for the variable.
Here's how it works:
  • Identify the coefficient of the variable; this is the number multiplying the variable.
  • Divide both sides of the equation by this coefficient to obtain the variable on one side.
  • Simplify to find the value of the variable.
The division method is simple and direct, making it a powerful tool for solving equations efficiently.

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Most popular questions from this chapter

Run the easy level of the LINES program on your calculator. [ \(\ \square\) See Calculator Note 3D to learn how to use LINES program. 4] Sketch a graph of the randomly generated line on your paper. Use the trace function to locate the \(y\)-intercept and to determine the rate of change. When the calculator says you have the correct equation, write it under the graph. Repeat this program until you get three correct equations in a row.

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